In Lab experiment
Oscar Feng-Hsun Chang
2017 May
Based on the classic paper by Holling in 1959, predator-prey interaction can be modeled by the following functions (i.e. so called Type I or Type II functional response). Both functional responses contains a parameter that describes the proportion of prey that is available for the predator. 1. Type I functional response:
\[f(R) = {\alpha}TR \], where \({\alpha}\) is the encounter probability, \(T\) is the searching time, and \(R\) is the prey density. Encounter probability (\({\alpha}\)) is the proportion of prey that is available for the predator.
- Type II functional response:
\[f(R) = \frac{{\alpha}TR}{1+{\alpha}hR}\], where \({\alpha}\) is again the encounter probability, \(T\) is the searching time, and \(h\) is the handling time.
In my lab experiment, I directly modified the \({\alpha}\) parameter because I used the screen mesh to modify the proportion of IG prey that is available to the IG predator. This lab experiment would thus be valid to verify my model predictions.
Population dynamics visualization
======= >>>>>>> origin/masterFirst visualize the population dynamics of the two protozoa species.
Blepharisma
Colpidium
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Now I take the hour 368, 414, 468, 486, and 535 to calculate the mean and standard error of two protozoa density in the six treatments (0%, 20%, 40%, 60%, 80%, and 100% encounter probability).
Model predictions
From the model I derive three major predictions.
1. IG prey
Model prediction
Here I extract the IG prey density at the equilibrium from the model. According to the model that used type I functinoal response to model intra-guild predation, I would expect the IG prey density to monotonically decrease with encounter probability.
IG prey density at equilibrium in the model
Experimental results
Here I calculate the mean IG prey (Colpidium) density across the six treatments (0%, 20%, 40%, 60%, 80% and 100% encounter probability)
However, for the IG prey, I would expect its density to decrease with the increase of intra-guild predation rate, which, in theory, should be proportional to the encounter probability between IG prey and IG predator.
Treatment check
In terms of IG predator density, I would expect it to be similar across treatments. From the pilot experiment (see below), I’ve observed that the density of IG predator (Blepharisma) is not affected by the occurrence of intra-guild predation.
Population dynamics from pilot experiment
Mean Blepharisma density across treatments.
From this plot, the only treatment that would be significantly higher than the other treatments are the 20% treatment.
Statistical check
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Now I take the final 5 data points (hour 414, 468, 486, 535, 581) to calculate the mean and standard error of two protozoa density in the six treatments (0%, 20%, 40%, 60%, 80%, and 100% enounter probability).
In terms of IG predator density, I would expect it to be similar across treatments. From the pilot experiment, the density of IG predator (Blepharisma) is not affected by the occurance of intra-guild predation.
Blepharisma density across treatments.
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## Kruskal-Wallis rank sum test
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## data: avg by as.factor(trmt.all)
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## Kruskal-Wallis chi-squared = 15.083, df = 5, p-value = 0.01001## Kruskal-Wallis rank sum test
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## data: x and group
## Kruskal-Wallis chi-squared = 15.4702, df = 5, p-value = 0.01
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## Kruskal-Wallis chi-squared = 14.371, df = 5, p-value = 0.01342## Kruskal-Wallis rank sum test
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## data: x and group
## Kruskal-Wallis chi-squared = 15.3381, df = 5, p-value = 0.01
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## Comparison of x by group
## (Benjamini-Hochberg)
## Col Mean-|
## Row Mean | B00 B02 B04 B06 B08
## ---------+-------------------------------------------------------
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## B02 | -2.928891
## | 0.0085*
## |
## B04 | 0.289273 3.218164
## | 0.4456 0.0097*
## |
## B06 | -0.415830 2.513061 -0.705103
## | 0.4235 0.0224* 0.3606
## |
## B08 | 0.271193 3.200084 -0.018079 0.687023
## | 0.4212 0.0052* 0.4928 0.3355
## |
## B10 | -1.229411 1.699480 -1.518684 -0.813580 -1.500604
## | 0.2052 0.1338 0.1611 0.3466 0.1430The density of 20% encounter probability treatment has significantly higher density than the other treatments (the p-value of Kruskal-Wallis test is 0.0100129) and the pair-wise comparison (Dunn’s test) also show significant difference between 20% treatment versus other treatments.
Mean Colpidium density across treatments.
However, for the IG prey, I would expect its density to decrease with the increase of intra-guild predation rate, which, in theory, should be proportional to the encounter probability between IG prey and IG predator.
From the plot, all treatments has lower density comparing to the 0% treatment, which is the treatment where two protozoa species engage in only competition. This suggests that Colpidium density would be lower as long as intra-guild predation occurs.
Statistical check
======= ## B02 | -2.725718 ## | 0.0241* ## | ## B04 | -0.126357 2.599360 ## | 0.4497 0.0233* ## | ## B06 | -0.252715 2.473002 -0.126357 ## | 0.4618 0.0251 0.4818 ## | ## B08 | 0.992811 3.718529 1.119169 1.245526 ## | 0.2673 0.0015* 0.2466 0.2281 ## | ## B10 | -0.703993 2.021724 -0.577635 -0.451277 -1.696804 ## | 0.3611 0.0648 0.3842 0.4074 0.1122The density of 20% encounter probability treatment has significantly higher density than the other treatments (the p-value of Kruskal-Wallis test is 0.013417) and the pair-wise comparison (Dunn’t test) also show significant difference between 20% treatment versus other treatments.
However, for the IG prey, I would expect its density to decrease with the increase of intra-guild predation rate, which, in theory, should be proportional to the encounter probabilty between IG prey and IG predator.
Colpidium
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## Kruskal-Wallis rank sum test
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## data: avg by as.factor(trmt.all)
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## Kruskal-Wallis chi-squared = 21.447, df = 5, p-value = 0.0006668## Kruskal-Wallis rank sum test
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## data: x and group
## Kruskal-Wallis chi-squared = 21.4466, df = 5, p-value = 0
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## Kruskal-Wallis chi-squared = 12.476, df = 5, p-value = 0.02882## Kruskal-Wallis rank sum test
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## data: x and group
## Kruskal-Wallis chi-squared = 12.476, df = 5, p-value = 0.03
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## Comparison of x by group
## (Benjamini-Hochberg)
## Col Mean-|
## Row Mean | C00 C02 C04 C06 C08
## ---------+-------------------------------------------------------
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## C02 | 2.551246
## | 0.0201*
## |
## C04 | 3.144142 0.592895
## | 0.0062* 0.3192
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## C06 | 3.701104 1.149857 0.556962
## | 0.0016* 0.1706 0.3094
## |
## C08 | 0.700694 -1.850552 -2.443447 -3.000409
## | 0.3022 0.0602 0.0182* 0.0067*
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## C10 | 1.221723 -1.329522 -1.922418 -2.479380 0.521029
## | 0.1664 0.1531 0.0584 0.0197* 0.3012For the Colpidium, We see that 0% encounter probability treatment tend to have higher density than the other treatments, although the multiple comparison results show that 40% and 60% encounter probability treatment are the only two treatments that has significantly lower density.
Testing hypothesis
According to the mathematical model, I hypothesize that bacteria consumption would first increase and than decrease with intra-guild predation rate (see below). Again, I used the encounter probability between the two protozoa species to represent the intra-guild predation rate.
Theoretical model prediction
Check initial bacteria density.
First I check if the initial bacteria density across treatments are the same.
======= ## C02 | 1.473747 ## | 0.1318 ## | ## C04 | 2.426291 0.952544 ## | 0.0572 0.2840 ## | ## C06 | 2.983439 1.509692 0.557148 ## | 0.0214* 0.1639 0.3331 ## | ## C08 | 0.682956 -0.790791 -1.743335 -2.300483 ## | 0.3091 0.2925 0.1219 0.0536 ## | ## C10 | 0.952544 -0.521203 -1.473747 -2.030895 0.269587 ## | 0.2556 0.3226 0.1506 0.0792 0.3937For the Colpidium, We see that 0% encounter probability treatment tend to have higher density than the other treatments, although the multiple comparision results show that 60% encounter probability treatment is the only treatment that has significantly lower density.
Now I check the bacteria density.
First I check if the initinal bacteria density across treatments are the same.
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## Kruskal-Wallis rank sum test
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## data: T0 by as.factor(gp)
## Kruskal-Wallis chi-squared = 8.8608, df = 6, p-value = 0.1816## Kruskal-Wallis rank sum test
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## data: x and group
## Kruskal-Wallis chi-squared = 8.8608, df = 6, p-value = 0.18
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## Comparison of x by group
## (Benjamini-Hochberg)
## Col Mean-|
## Row Mean | 00 02 04 06 08 10
## ---------+------------------------------------------------------------------
## 02 | -0.285798
## | 0.4069
## |
## 04 | -1.460746 -1.174948
## | 0.2521 0.2800
## |
## 06 | -1.901142 -1.631688 -0.523936
## | 0.2005 0.2697 0.3940
## |
## 08 | -2.381652 -2.095854 -0.920905 -0.344301
## | 0.1810 0.1895 0.3125 0.4038
## |
## 10 | -1.016171 -0.730373 0.444575 0.943086 1.365480
## | 0.3250 0.3489 0.4056 0.3299 0.2582
## |
## ctrl | -1.556013 -1.270214 -0.095266 0.434119 0.825639 -0.539841
## | 0.2514 0.2678 0.4621 0.3875 0.3304 0.4125The initial bacteria density are not significantly different from each other.
<<<<<<< HEADTesting hypothesis
Now I check the bacteria consumption in different treatment. Bacteria consumption is calculated by the difference between density at the stable state (\(T_{end}\)) subtracting the initial bacteria density (\(T_0\)).
Now I check the bacteria consumption in different treatment. Bacteria consumption is calculated by the difference between density at the stable state (\(T_{end}\)) substracting the initial bacteria density (\(T_0\)).
mod_bac = kruskal.test(dif~as.factor(gp), data=Bac_pop_stat)
mod_bac##
## Kruskal-Wallis rank sum test
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## data: dif by as.factor(gp)
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## Kruskal-Wallis chi-squared = 21.335, df = 6, p-value = 0.001597summary(mod_bac)## Length Class Mode
## statistic 1 -none- numeric
## parameter 1 -none- numeric
## p.value 1 -none- numeric
## method 1 -none- character
## data.name 1 -none- characterdunn.test(Bac_pop_stat[,"dif"],
Bac_pop_stat[,"gp"], method="bh")## Kruskal-Wallis rank sum test
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## data: x and group
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## Kruskal-Wallis chi-squared = 19.439, df = 6, p-value = 0
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## Kruskal-Wallis chi-squared = 8.6294, df = 6, p-value = 0.2
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##
## Comparison of x by group
## (Benjamini-Hochberg)
## Col Mean-|
## Row Mean | 00 02 04 06 08 10
## ---------+------------------------------------------------------------------
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## 02 | -2.095854
## | 0.0421
## |
## 04 | -0.952661 1.143193
## | 0.2385 0.2213
## |
## 06 | 0.209574 2.185565 1.107752
## | 0.4378 0.0433 0.2164
## |
## 08 | -2.222875 -0.127021 -1.270214 -2.305322
## | 0.0459 0.4495 0.1947 0.0444
## |
## 10 | -3.048515 -0.952661 -2.095854 -3.083742 -0.825639
## | 0.0121* 0.2556 0.0474 0.0215* 0.2684
## |
## ctrl | -2.762716 -0.666862 -1.810055 -2.814289 -0.539841 0.285798
## | 0.0150* 0.3118 0.0738 0.0171* 0.3438 0.4283Bac_pop_stat[,"gp"] = relevel(as.factor(Bac_pop_stat[,"gp"]), ref="ctrl")
mod_bac1 = lmer(dif~as.factor(gp)+(1|rep), data=Bac_pop_stat, REML=FALSE)
coefs = data.frame(coef(summary(mod_bac1)))
# use normal distribution to approximate p-value
coefs[,"p.z"] = 2 * (1 - pnorm(abs(coefs$t.value)))
coefs## Estimate Std..Error t.value p.z
## (Intercept) 248.431122 95.56593 2.59957830 9.333838e-03
## as.factor(gp)00 -334.789539 135.15064 -2.47715843 1.324331e-02
## as.factor(gp)02 -152.652633 135.15064 -1.12949992 2.586870e-01
## as.factor(gp)04 -686.669790 135.15064 -5.08077363 3.759008e-07
## as.factor(gp)06 -578.201915 143.34890 -4.03352886 5.494543e-05
## as.factor(gp)08 24.343490 135.15064 0.18012116 8.570574e-01
## as.factor(gp)10 8.625465 135.15064 0.06382112 9.491127e-01## Estimate Std..Error t.value p.z
## (Intercept) 214.58945 131.3286 1.6339891 0.1022611360
## as.factor(gp)00 -188.27475 185.7266 -1.0137197 0.3107165232
## as.factor(gp)02 -28.78787 185.7266 -0.1550013 0.8768203045
## as.factor(gp)04 -667.10023 185.7266 -3.5918392 0.0003283525
## as.factor(gp)06 -606.56531 196.9929 -3.0791235 0.0020761062
## as.factor(gp)08 -71.92755 185.7266 -0.3872764 0.6985515704
## as.factor(gp)10 67.76148 185.7266 0.3648452 0.7152269367